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Course Overview

This course is the equivalent of an introductory college-level calculus course. Calculus helps scientists, engineers, and financial analysts understand the complex relationships behind real-world phenomena. Students learn to evaluate the soundness of proposed solutions and apply mathematical reasoning to real-world models. Students also learn to understand change geometrically and visually (by studying graphs of curves), analytically (by studying and working with mathematical formulas), numerically (by seeing patterns in sets of numbers), and verbally. Students prepare for the AP Exam and further studies in science, engineering, and mathematics.

Course Outline

SEMESTER ONE

Unit 1: The Basics

Students prepare to study calculus by reviewing some basic precalculus concepts from algebra and trigonometry. They learn what calculus is, why it was invented, and what it's used for.

Pre-Calculus Review

Introduction to Calculus

Using a Graphing Calculator

Combining Functions

Composite and Inverse Functions

Graphical Symmetry

Patterns in Graphs

Unit 2: Applications of the Integral

The topics referred to below are those listed in the College Board's Calculus AB topic outline. This unit addresses Topic I: Functions, Graphs, and Limits. Students learn two important concepts that underlie all of calculus: limits and continuity. Limits help students understand differentiation (the slope of a curve) and integration (the area inside a curved shape). Continuity is an important property of functions.

Finding Limits Analytically

Asymptotes and Limits

Relative Magnitudes for Limits

When Limits Do and Don't Exist

Continuity

Intermediate and Extreme Value Theorems

Unit 3: The Derivative

This unit addresses Topic II: Derivatives. Students learn how to calculate a derivative, the slope of a curve at a specific point. They learn techniques for finding derivatives of algebraic functions (such as y = x2) and trigonometric functions (such as y = sin x).

Derivatives at a Point

The Derivative

The Power Rule

Sums, Differences, Products, and Quotients

Graphs of Functions and Derivatives

Continuity and Differentiability

Rolle's and Mean Value Theorems

Higher-Order Derivatives

Concavity

Chain Rule

Implicit Differentiation

Unit 4: Rates of Change

This unit focuses on Second Derivatives and Applications of Derivatives within Topic II: Derivatives. Students learn how to use calculus to model and analyze changing aspects of our world.

Extrema

Optimization

Tangent and Normal Lines

Tangent Line Approximation

Rates and Derivatives

Related Rates

Rectilinear Motion

Unit 5: Semester Review and Test

Students review what they have learned so far and take the semester exam.

Semester 1 Review

Semester 1 Exam

SEMESTER TWO

Unit 1: The Integral

This unit focuses on Topic III: Integrals. Students learn numerical approximations to definite integrals, interpretations and properties of definite integrals, the Fundamental Theorem of Calculus, and techniques of antidifferentiation. They learn how to find areas of curved shapes.

Riemann Sums

Area Approximations

The Definite Integral

Properties of Integrals

Graphing Calculator Integration

Applications of Accumulated Change

Antiderivatives

Composite Functions

The Fundamental Theorem of Calculus

Definite Integrals of Composite Functions

Analyzing Functions and Integrals

Unit 2: Applications of the Integral

This unit focuses on Topic III: Integrals. Students learn to use integrals and antiderivatives to solve problems.

Area

Volumes of Revolution

Cross Sections

More Rectilinear Motion

Other Applications of the Definite Integral

Unit 3: Inverse and Transcendental Functions

This unit focuses on Topic II: Derivatives and Topic III: Integrals. Students learn to calculate and use derivatives, antiderivatives, and integrals of exponential functions (such as y = 3x where the input variable is an exponent), logarithmic functions (the inverses of exponential functions), and inverse trigonometric functions (such as y = secant(x)).

Inverse Trig Functions

Review of Logarithmic and Exponential Functions

Transcendentals and 1/x

Derivatives of Logs and Exponentials

Analysis of Transcendental Curves

Integrating Transcendental Functions

Applications of Transcendental Integrals

Unit 4: Separable Differential Equations and Slope Fields

This unit focuses on Topic II: Derivatives; specifically, on Equations Involving Derivatives. Students investigate differential equations, and solve the equations using a technique called "separating the variables."

Slope Fields

Differential Equations as Models

Exponential Growth and Decay

More Applications of Differential Equations

Unit 5: AP Exam Review and Final Exam

Students review what they have learned and prepare for the AP Exam with practice tests that simulate the AP test experience.

Calculus as a Cohesive Whole

Review of Topics

Practice Final Exams

Unit 6: Calculus Project

Teachers may choose to assign a final project.

Project Days

Additional Information

Course Length

4 Months

Prerequisites

No

Course Materials

No

Course Start Date

Courses Taught by a K12 Teacher

Courses with a teacher have designated start dates throughout Fall, Spring, and Summer. Full-year courses last 10 months and semester courses last 4 months. Courses are taught by teachers in K12 International Academy. For details on start dates, click here.

Teacher Assisted

Yes, this course is taught by a K12 International Academy teacher.
If you are looking for a teacher-supported option with additional flexibility and year-round start dates, click here to learn about the Keystone School, another K12 online private schooling option.

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